That blacker-than-black material on the table and his shirt hide a lot. The coins are always in the exact same spot when they "move" from location to location. He's got rotating "holes". A mechanical device somehow holds on to the coins (probably a well-timed and rehearsed electro-magnet) and spins to pull the coin into the hole and replaces it with that same black fabric. It only needs to spin faster then a 60th of a second to not be caught by the video camera. The table wobbles like mad when ever the coins "disappear". It's the spinning device. Picture a small paddle with the coin on one side and black fabric on the other. It is spring loaded to pull down into the table and then spin and then pop back into place. It is precisely engineered to not disturb the cards on the popup.

He then moves the coins to new holes pre-loaded with plastic flower petals that pop up when the coins are flipped into those holes. The little doors are flipping counter-clockwise from our vantage point and you can see the petals move slightly left with momentum.

The disappearing cards are simple dropped into his lap. that black shirt hides whatever caught them.

The coins did spin on the table and they were too fast for human eyes to spot, but the video camera caught the spin. The producers cheated and freeze framed some of the spins to disguise them. This, I would suppose, was to give the home viewers the same experience that the live audience had.